Observing Dark Worlds: A Beginners Guide to Dark Matter & How to Find It

Here at Kaggle we are very excited to launch a brand new Kaggle Recruit competition: Observing Dark Worlds (ODW). Being an Astrophysicist as well as a great lover of everything weird and wonderful such a competition really gets my motors going.

The subject of Dark Matter is commonly grouped with similar abstract concepts such as aliens, black holes, supernovae and the big bang, assumed to be incomprehensible and inaccessible. However, speaking from personal experience, grasping Dark Matter needn't require more than a wine glass and a candle. One of the main aims of ODW is to open up the metaphorical Dark Matter doors to all, engaging every type of data scientist, with the added hope of helping to explain the Universe!

In order to grasp the concept of Observing Dark Worlds, it would be good to get an understanding of what Dark Matter actually is, why we are interested in it and what we hope to achieve.

What is Dark Matter?

Dark Matter is a particle (or composed of particles), like electrons, quarks, protons, but fundamentally different in the way it interacts. What type of particle it is, is yet to be seen, but evidence suggests that it could be part of some completely new family of particles.

It is thought that in the beginning the distribution of Dark Matter throughout the Universe was pretty uniform, however as small bits drew themselves together through the force of gravity, it started to clump and aggregate. As larger bodies coalesced it started to form huge structures leading to the present day where it exists as a cosmic spider's web, with strand like filaments funneling matter down to connecting points where you find huge clumps called halos. More abundant than everything we can see by a factor of 7:1, Dark Matter acts like a cosmological scaffolding. It provides the basic framework for all visible matter to form on. Its gravitational pull determines the positions of the galaxies and stars in the Universe, and in the environments of the most massive halos we see lots of galaxies all bound together orbiting one another.

Observed distribution of galaxies in the sky. Each dot is a galaxy. It can be seen how the galaxies reveal the web like structure in the Universe with filaments and halos. Credit: M. Blanton & SDSS Collaboration, www.sdss.org

So how do we go about finding Dark Matter? As you may quite reasonable ask, how do we see something that is dark? And that's the crux of it: the problem with Dark Matter is that, well, its dark and we can't see it. Fortunately something so vast and massive does not go unnoticed. As Dark Matter halos become increasingly large they actually bend the fabric of spacetime, just as when you put a golf ball on a sheet of rubber it bends and distorts the rubber. If you add more golf balls the rubber sheet will become increasingly distorted. The result is that anything that passes close enough to the golf ball will 'fall' in, and become affected by the distorted space it is travelling in. If the object is massive enough it will fall in and be consumed by the object, i.e. a black hole, however if there is only a slight bending of spacetime the paths of particles moving past it will only be slightly bent. If we imagine a scenario in which such a passing particle is in fact a light particle (or photon), as it experiences the deformed spacetime it will roll down slightly and come out of the distortion at a different angle to when it went in. This results in the object which emitted the photon appearing different to us than it would in the absence of the deformation. In space the emitting objects of interest are galaxies and since there are a lot of galaxies behind these halos of dark matter that are causing spacetime distortions, all the photons and hence the shapes of the galaxies will appear different in a way that reflects the position of the Dark Matter.

Dark Matter has proven to be very elusive. We have been trying to smash stuff together to find it (LHC CERN), we have tanks of xenon in abandoned mines trying to detect a signal of one interaction a year, yet still we have no definitive evidence for dark matter on Earth. The only place that we currently have direct evidence for Dark Matter is from the motion and behaviour of the galaxies and stars in the Universe (e.g The Bullet Cluster). Therefore the only way we can get an insight into the properties of Dark Matter is through the study of the wider Universe. If we can pin down what Dark Matter is from our observations maybe we can help with focus of Earth studies.

A huge Dark Matter halo has pulled together a group of galaxies. The vast amount of Dark Matter has caused the image of a background galaxy to look like a smeared arc across the image. Credit: NASA, ESA, J. Richard (CRAL) and J.-P. Kneib (LAM). Acknowledgement: Marc Postman (STScI)

Understanding the properties of Dark Matter requires accurate estimates of its peak position. If we can nail down the peak of the density profile for a halo we can explore its properties. However estimating this both accurately and precisely is extremely difficult.

The aim of ODW is to develop an algorithm that can pin point the positions of Dark Matter halos extremely precisely and without directional bias . If we can do this then we can fully exploit future space missions such as Euclid.

How the competition is structured

So the aim of ODW is to develop an algorithm that can predict the position of Dark Matter halos. What we have done is simulate a number of skies filled with galaxies. Each sky will contain between 300 and 700 galaxies with a x and y sky coordinate between 0 and 4200. We will then place either 1, 2 or 3 Dark Matter halos in the sky between the galaxies and us such that the galaxies are distorted. We will then tell you what the ellipticities of the galaxies are. Ellipticity can be split into two components: e1 and e2. e1 describes the elongation of a galaxy in the x direction (positive e1) and the y direction (negative e1). e2 describes the elongation of a galaxy in the 45 degree angle. So positive e2 is a galaxy elongated on the 45 degree line and negative e2 is elongation in the 135 angle. For more on ellipticity please see the Introduction to Ellipticity page. So the the test_sky and train_sky data will look for example like this (except in csv format):

GalaxyId

x

y

e1

e2

Galaxy1

1234.56

4000.21

0.1422

-0.03212

Galaxy2

3214.23

1232.32

0.3444

0.0233

…

…

…

…

…

We will then provide you with the answers to the training data, which will look like this (again in csv format)

SkyId

N

x_ref

y_ref

halo_x1

halo_y1

halo_x2

halo_y2

halo_y3

halo_y3

Sky1

1

1086.8

1114.61

1086.8

1114.61

0

0

0

0

…

…

…

…

…

…

…

…

…

…

Sky142

2

3477.71

1907.33

3477.71

1907.33

232.32

4001.11

0

0

…

…

…

…

…

…

…

…

…

…

Sky223

3

2315.78

1081.95

2315.78

1081.95

2312.32

2981.24

198.23

3889.01

…

…

…

…

…

…

…

…

…

…

Where the first column is the ID of the sky and allows you to cross reference the data file in train_skies and the answer in Training_halos.csv, then the number of halos in that sky, a x and y reference point which is used to calculate the metric, and then the true x and y coordinate of each halo. In the case when a halo is not present (i.e. there is only one or two halos in the sky) the position of the halo will be 0. Note: In the file the header for column 'N' will in fact be named numberHalos. We will also give you a file called 'testhalosCount.csv' which will have the information of the number of halos in each sky in the test set.

Observing Dark Worlds: Getting you started
The data page presents two benchmarks in python which have been used to calculate the positions of halos. Although similar they are in fact calculating two different things. One is model dependant, i.e. it assumes some kind of relationship between the force of gravity and the distance the galaxy is from the peak of the Dark Matter density, the other just calculates the signal in a gridded image. Here are some snippets of code to help you on your way to calculating the positions of a Dark Matter halos.

Benchmark #1: Creating a Signal Map

def dark_matter_finder( x_galaxy, y_galaxy, e1, e2, x_halo, y_halo)
"""Function to calculate the Dark Matter signal around a proposed position
Arguments :
x_galaxy, y_galaxy: Vectors containing the x and y coordinate of each galaxy in the sky
e1, e2: The 2 components of ellipticity for each galaxy in the sky
x_halo, y_halo: The estimated coordinates of the halo
Returns :
signal : Scalar value of the total signal given the proposed halo
"""
# Find out the angle each galaxy is at with respects to my guessed position of the halo
angle_wrt_halo = arctan((y_galaxy-y_halo)/(x_galaxy-x_halo))
# Calculate the total signal for a halo at my guessed position
signal = sum( -(e1*np.cos(2.0*angle_wrt_halo) + \ e2*np.sin(2.0*angle_wrt_halo)) )
return signal

Once you have calculated this you can search the image for the spots with the highest signal

if __name__ == "__main__":
""" Main program to determine the position of a halo """
# Read in the data from the Sky test file
x_galaxy, y_galaxy, e1, e2 = loadtxt('Test_Sky1.csv', usecols=(1, 2, 3, 4),\ skiprows=1, unpack=True)
#I want to search the sky in a grid like fashion, so I want to split # the skyup and find the signal at each point in the grid
Number_of_bins = 10
Sky_size = 4200.0
#It is square in all cases
Binwidth = Sky_size/float(Number_of_bins)
gridded_map= zeros([Number_of_bins, Number_of_bins], float)
for i xrange(Number_of_bins):
for j in xrange(Number_of_bins):
x_halo = i*binwidth # Proposed x position of the halo
y_halo = j*binwidth # Proposed y position of the halo
gridded_map[i,j] = dark_matter_finder(x_galaxy, y_galaxy, e1, e2,\ x_halo, y_halo)
estimated_x_position_halo=where(signal == max(gridded_map))[0][0]*binwidth
estimated_y_position_halo=where(signal == max(gridded_map))[1][0]*binwidth

This code would work reasonably well in the case of one halo. It would need to be extended if there were more than one.

Benchmark #2: Maximum Likelihood
So another option, instead of finding the signal at a given point, it is possible to assume a profile of Dark Matter halo and then try to fit this model to the data. From this find the most likely position of the halo. So one such model could be that the distortion caused by a Dark Matter halo has a 1/r drop off, where r is the distance from the center of the halo. This code finds the likelihood of a halo at a particular position and then assumes that the position with the maximum likelihood is the position of the halo. So using the same main function as before but re-defining the dark_matter_finder function:

def dark_matter_finder( x_galaxy, y_galaxy, e1, e2, x_halo, y_halo) :
"""Function to calculate the likelihood of a dark matter halo given a proposed position
Arguments:
x_galaxy, y_galaxy: Vectors containing the x and y coordinate of each galaxy in the sky
e1, e2: The 2 components of ellipticity for each galaxy in the sky
x_halo, y_halo: The estimated coordinates of the halo
Returns :
likelihood : Scalar value of the likelihood given the proposed halo
"""
# Find out the radial distance and angle each galaxy is at with respects to my guessed position of the halo
radial_distance_galaxy_from_halo = sqrt( (x_galaxy-x_halo)**2 +\ (y_galaxy-y_halo)^2 )
angle_wrt_halo = arctan((y_galaxy-y_halo)/(x_galaxy-x_halo))
#We assume that the ellipticity caused by this is 1/r
total_ellipticity = 1.0/radial_distancee_galaxy_from_halo
#Then convert this into the two components of ellipticity, e1 and e2
e1_model = -total_ellipticity*cos(2.0*angle_wrt_halo)
e2_model = -total_ellipticity*sin(2.0*angle_wrt_halo)
#Now work out the chi-square fit of the model with compared to the data
chi_square_fit = sum( (e1_model - e1)**2 + (e2_model - e2)**2 )
#Convert to likelihood
likelihood = exp((-chi_square_fit/2.0))
return likelihood

Once again this snippet of code will only work for one halo. It is possible to simultaneously fit two halos (or three) to the sky so that you are fitting more parameters, for example by adding the effects of multiple halos.

Although we have supplied you with two introductory methods please feel free to explore alternative routes (in fact we encourage it!) or use online code. One benchmark we have provided is a public code called LENSTOOL. LENSTOOL fits realistic models of dark matter halos to data using a clever sampling technique. It is one of the leading algorithms in this work you are more than welcome to build upon it!

The Observing Dark Worlds Metric: How are we testing you and how are you going to win?!

So the aim of the competition is to create an algorithm that can recognise the features of a Dark Matter halo in a field of galaxies. In future experiments we want to exploit entire data sets and therefore it is imperative that there is no residual bias in the algorithm, however we also don't want to lose sight of the main goal which is estimating the positions as accurately as possible. To this affect we have created a metric which will sum the average distance your estimates are away from the true position and the average angular vector of your position. (These will be weighted such they are both dimensionless and similar orders of magnitude.) For more details on the metric please visit the evaluation page of the competition.

Is my algorithm really going to change the Universe?

The business of Gravitational Lensing (the bending of light due to matter) is a relatively new one, and therefore an opportunity exists to really change the way we approach this problem. We are looking for innovative, new and thought provoking methods that can solve this issue in a manner that the astrophysics community had never thought of.

The Wider Picture

Observing Dark Worlds

Observing Dark Worlds (ODW) will be an advert to not only astronomers but all investment banks and financial institutions. It will not only provide new ways to reconstruct the positions of Dark Matter but it will show how you can use abstract concepts as a method for recruitment filtering. Companies spend billions each year on finding the best of the best. Interviews, assessment centres and recruitment days, all of which cost money. Kaggle recruit allow companies to already filter out the best from the rest. By implementing difficult astronomical questions you are testing a participant’s ability to apply their knowledge to different situations and scenarios. It allows those scientists out there who may have credentials that would normally get filtered out at the first hurdle to be considered for jobs. Kaggle have already helped major companies recruit new staff, who they admit, would never have considered purely on their academic background. Winton Capital, one of the largest hedge funds in the world, is sponsoring the ODW competition. They are looking to offer a budding data scientist a potential career by putting up the prize money. This is a great example of how the money to sponsor competitions doesn’t always have to come from within the community.. They are looking for the brightest minds the world has to offer and all you have to do is grab their attention by solving the problems of the Universe. Good Luck and Happy Hunting!

There are a large number of typos in the code above. No big deal, just a heads-up. I can supply corrected code if the authors want to publish more accurate code - or everybody can just do what I did. Again, no big deal - it only took me 3 minutes to clean up.